title: “Statistical methods for movement ecology” author: Marie-Pierre Etienne institute: "" date: “2022” csl: “../resources/apa-no-doi-no-issue.csl” output: xaringan::moon_reader: css: [‘metropolis’, ‘mpe_pres_HDR.css’] lib_dir: libs nature: ratio: 16:10 highlightStyle: github highlightLines: true countIncrementalSlides: false beforeInit: ‘../courses_tools/resources/collapseoutput.js’ — name: intro
.legend[ Movement drivers by Nathan, Getz, Revilla, et al. (2008) ] ]
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.question[Movement informs on internal states and habitat preferences]
| name: move2data # From Movement to Movement data |
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.pull-right[ A continuous process sampled at some discrete potentially irregular times.
Time series with values in \(\mathbb{R}^2\) (on earth …).
\[\begin{array}\\ \mbox{Time} & \mbox{Location} & \mbox{Turning angle} & \mbox{Speed}\\ t_{0} & (x_0, y_0) & NA & NA\\ t_{1} & (x_1, y_1) & NA & sp_1\\ t_{2} & (x_2, y_2) & ang_2 & sp_2\\ \vdots & \vdots& \vdots& \vdots \\ t_{n} & (x_n, y_n) & ang_n & sp_n\\\\ \end{array}\] ]
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| name: data2model # From Movement Data to Movement Model |
Often analysed using discrete time model (McClintock, Johnson, Hooten, et al., 2014)
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.pull-right[ ## Accounting for internal states
Classically addressed with Hidden Markov Model
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Let \(\boldsymbol{\tau}={\tau_1,...,\tau_{K-1}}\) ( \(\tau_0=-1\) ) be a partition in K segments of \(\{1,\ldots n\}\) \[\boldsymbol{X}_{i}\overset{i.i.d}{\sim}\mathcal{L}(\theta_k),\quad \forall i \in \{ \tau_{k-1}+1:\tau_k \}\] The .care[Dynamic Programming algorithm] allows to explore efficiently all possible segmentation and to estimate \(\boldsymbol{\hat{\tau}}\) ]
– .pull-right[Let \(Z_k\) stand for the class of segment \(k,\) \(\forall i \in \{ \tau_{k-1}+1:\tau_k \}\) \[Z_k \overset{i.i.d}{\sim} \mathcal{M}(\pi), \quad\boldsymbol{X}_{i}\vert Z_k=l \overset{i.i.d}{\sim}\mathcal{L}(\theta_l)\] The Dynamic Programming coupled with EM algorithm allows to explore efficiently all possible segmentation and to estimate \(\boldsymbol{\hat{\tau}}\) ]
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In (Patin, Etienne, Lebarbier, et al., 2019) : a direct extension to simultaneous segmentation for home range shift.
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.question[Movement path is more than time series, importance of considering the space.]
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.center[.care[Proposing ecologically meaningful movement models]]
Pros and cons of Discrete time versus continuous time movement models discussed in McClintock, Johnson, Hooten, et al. (2014)
Let \((X_s)_{s\geq0}\in\mathbb{R}^2\) denote the position at time \(s\).
.pull-left[ * Brownian motion: a pure diffusion model \[dX_s = dW_s, \quad X_0=x_0.\]]
– .pull-left[ * Ornstein Uhlenbeck process: central place behavior \[dX_s = -B (X_s- \mu) ds + dW_s, \quad X_0=x_0.\]
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Popular models as Brownian Motion and Ornstein Uhlenbeck have known transition densities \(q(x_t, x_{t+s})\) which is not the case in general.
Brillinger, Preisler, Ager, et al. (2002) propose a flexible framework \[dX_s = -\nabla H(X_s) ds + \gamma dW_s, \quad X_0=x_0.\]
but no explicit transitions \(q(x_t, x_{t+s})\)
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In Gloaguen, Etienne, and Le Corff (2018a), as part of P. Gloaguen’s PhD, explore \(H(X_s) = \sum_{k=1}^K \pi_k \varphi_k(X_s),\)
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.pull-right[ * Euler approximation : biased estimates with low frequency data * (Ozaki, 1992) and Kessler (1997) same results than * MCEM based on exact simulation (Beskos, Papaspiliopoulos, Roberts, et al., 2006) limits the flexibility of the SDE.]
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.pull-rights[ Let \(Y_k\) be the recorded position \(s_k\), a noisy observation of the true position \(X_k\):
\[dX_s = b(X_s) ds + \gamma dW_s, \quad X_0=x_0; \quad Y_k \overset{ind}{\sim} \mathcal{L} (X_k, \theta{o}).\] ]
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\[\sum_{k=0}^{n-1}\mathbb{E}( h(X_k, X_{k+1}) \vert Y_{0:n})\]
| * The particle-based, rapid incremental smoother (PaRIS) algorithm (Olsson and Westerborn, 2017) provides an online smoother using a rewriting of the Backward weight and an acceptation/rejection mechanism but depends on \(q(\xi_{k-1}, \xi_{k})\) |
| * The generalized random PaRIS algorithm, in (Gloaguen, Etienne, and Le Corff, 2018b), uses simple Euler approximation to propose the particles and uses a General Poisson Estimator to replace \(q(\xi_{k-1}, \xi_{k})\) with an unbiased estimator. |
.care[Restrictive constraints on the drift and the diffusion term], are relaxed in (Martin, Etienne, Gloaguen, et al., 2021).
A classical choice of resource selection function, (i.e stationary distribution including covariates) \[\pi{\left(x \vert \beta\right)} \varpropto \exp\left(\sum_{j=1}^J \beta_j c_j (x) \right). \]
Diffusion, under regularity condition admits a stationary distribution.
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Combining the ideas of (Michelot, Blackwell, and Matthiopoulos, 2019), and (Brillinger, Preisler, Ager, et al., 2002) lead to the Langevin diffusion as movement model, \[ d X_t = \frac{\gamma^2}{2} \nabla \log \pi{\left(X_t\right)} \, d t + \gamma \,d W_t,\quad X_0 =x_0. \] –
Using Euler approximation
\[ X_{i+1} \vert \lbrace X_i = x_i \rbrace = x_i + \frac{\gamma^2 \Delta_i}{2} \sum_{j=1}^J \beta_j \nabla c_j(x_i) + \sqrt{\Delta_i} \varepsilon_{i+1},\quad \varepsilon_{i+1} \overset{ind}{\sim} N \left( {0} , \gamma^2 \boldsymbol{I}_d \right),\]
leads to a simple linear model published in Michelot, Etienne, Blackwell, et al. (2019).
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Lejay and Pigato (2018) define a threshold diffusion and proposes an ML estimation method. Explore the generalisation to \(\mathbb{R}^2\).
| # A few words to finish |
| * Close interaction with biologist, - it’s helpful, - provides exciting statistical problems, - experiment a large diversity of approaches. * Hidden variables approach - provide flexible models (spatial abundance, classification in time series, Partially observed SDE) - popularized in Ecology in a Bayesian setting but frequentist approach is also possible * The Markovian property - not so realistic, - but a key component in both theoretical approach and computational aspects |
class: biblio # Bibliography
Beskos, A., O. Papaspiliopoulos, G. O. Roberts, et al. (2006). “Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion)”. In: Journal of the Royal Statistical Society: Series B (Statistical Methodology) 68.3, pp. 333-382.
Brillinger, D. R., H. K. Preisler, A. A. Ager, et al. (2002). “Employing stochastic differential equations to model wildlife motion”. In: Bulletin of the Brazilian Mathematical Society 33.3, pp. 385-408.
Gloaguen, P., M. Etienne, and S. Le Corff (2018b). “Online sequential Monte Carlo smoother for partially observed diffusion processes”. In: EURASIP Journal on Advances in Signal Processing 2018.1, p. 9.
Gloaguen, P., M. Etienne, and S. Le Corff (2018a). “Stochastic differential equation based on a multimodal potential to model movement data in ecology”. In: Journal of the Royal Statistical Society: Series C (Applied Statistics) 67.3, pp. 599-619. DOI: 10.1111/rssc.12251.
Kessler, M. (1997). “Estimation of an ergodic diffusion from discrete observations”. In: Scandinavian Journal of Statistics 24.2, pp. 211-229.
Lavielle, M. (2005). “Using penalized contrasts for the change-point problem”. In: Signal processing 85.8, pp. 1501-1510.
Lejay, A. and P. Pigato (2018). “Maximum likelihood drift estimation for a threshold diffusion”. In: Scandinavian Journal of Statistics.
| class: biblio count: false # Bibliography |
| Martin, A., M. Etienne, P. Gloaguen, et al. (2021). “Backward importance sampling for online estimation of state space models”. working paper or preprint. URL: https://hal.archives-ouvertes.fr/hal-02476102. |
| McClintock, B. T., D. S. Johnson, M. B. Hooten, et al. (2014). “When to be discrete: the importance of time formulation in understanding animal movement”. In: Movement Ecology 2.1, p. 21. |
| Michelot, T., P. G. Blackwell, and J. Matthiopoulos (2019). “Linking resource selection and step selection models for habitat preferences in animals”. In: Ecology 100.1. |
| Michelot, T., M. Etienne, P. Blackwell, et al. (2019). “The Langevin diffusion as a continuous-time model of animal movement and habitat selection”. In: Methods in Ecology and Evolution. |
| Nathan, R., W. M. Getz, E. Revilla, et al. (2008). “A movement ecology paradigm for unifying organismal movement research”. In: Proceedings of the National Academy of Sciences 105.49, pp. 19052-19059. |
| Olsson, J. and J. Westerborn (2017). “Efficient particle-based online smoothing in general hidden Markov models: the PaRIS algorithm”. In: Bernoulli 23.3, pp. 1951-1996. |
| Ozaki, T. (1992). “A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach”. In: Statistica Sinica, pp. 113-135. |
class: biblio count: false # Bibliography
Patin, R., M. Etienne, E. Lebarbier, et al. (2019). “Identifying stationary phases in multivariate time series for highlighting behavioural modes and home range settlements”. In: Journal of Animal Ecology.
Picard, F., S. Robin, E. Lebarbier, et al. (2007). “A segmentation/clustering model for the analysis of array CGH data”. In: Biometrics 63.3, pp. 758-766.